Homology and cohomology of compact vs non compact spaces

Question

What is special about the homology of the compact spaces, what are the most elementary properties that are verified by homology of compact spaces that are not true for the homology of non compact spaces ? I'm asking because i read that Euler characteristic is not easily defined for non compact spaces while it is straightforward for compact spaces and i also read that we have a homology theory that is better suited for noncompact spaces rather than singular homology, what is making this difference between compact and noncompact spaces regarding homology and cohomology ? Thank you for your help !

Answer 1

There are essentially no special properties of (singular) homology and cohomology of compact spaces, in general. A concrete result along these lines is that every topological space which is not too large (where "too large" is so large you cannot even prove such a large set exists in ZFC) is weak homotopy equivalent to a compact Hausdorff space. Since a weak homotopy equivalence induces isomorphisms on singular (co)homology, this means anything you that can happen with them in general spaces can also happen in compact Hausdorff spaces (as long as your spaces aren't ridiculously huge). To be precise, "not too large" means "smaller than the least measurable cardinal". This is a theorem of Adam Przeździecki; see this MO post and the comments under it.

When you restrict to sufficiently nice spaces, though, (co)homology of compact spaces tends to be better behaved since it is guaranteed to be finitely generated. For instance, if your space is a CW-complex, then it is compact iff it has only finitely many cells, and if it has only finitely many cells its (co)homology will be finitely generated since the cellular (co)chain complex is finitely generated. This finiteness is required, for instance, in order for the usual definition of Euler characteristic to make sense.

Another difference is that sometimes you are interested not in ordinary cohomology but in cohomology with compact support (or other related variations), which arises naturally in studying things like Poincaré duality (in the context of de Rham cohomology, for instance, it arises naturally because you can integrate a form over a compact set but over a non-compact set the integral may diverge). For compact spaces, cohomology with compact support is the same thing as ordinary cohomology, so you don't need to make a distinction between them.